Pitch space: Encyclopedia II - Pitch space - History of pitch space

Pitch space - History of pitch space

The idea of pitch space goes back at least as far as the ancient Greek music theorists known as the Harmonists. To quote one of their number, Bacchius, "And what is a diagram? A representation of a musical system. And we use a diagram so that, for students of the subject, matters which are hard to grasp with the hearing may appear before their eyes." (Bacchius, in Franklin, Diatonic Music in Ancient Greece.) The Harmonists drew geometrical pictures so that the intervals of various scales could be compared visually; they thereby located the intervals in a pitch space.

Cognitive psychologists including Longuet-Higgins (1978) and Shepard (1982), and composers and theorists including Weber (1824), Riemann, and Schoenberg (1954) created models of pitch space, modulatory space, or chordal space. For pitch space there are generally at least two dimensions, one for pitch class and one for register (i.e., the specific pitch), but there may be any number. (Lerdahl, 1992)

M.W. Drobisch (1855) was the first to suggest a helix (i.e. the spiral of fifths) to represent octave equivalency and reoccurrence (Lerdahl, 2001), and hence to give a model of pitch space. Shepard (1982) uses a double helix of two wholetone scales over a circle of fifths which he calls the "melodic map" (Lerdahl, 2001). Michael Tenzer suggests its use for Balinese gamelan music since the octaves are not 2:1 and thus there is even less octave equivalency than in western tonal music (Tenzer, 2000). See also chromatic circle.

The use of a lattice was first proposed by Euler (1739) to model just intonation using an axis of perfect fifths and another of major thirds (Lerdahl, 2001). James Tenney argues for multidimensional lattices, especially for just intonation systems, which contain a dimension for every pitch axis used (Tenney, 1983). Thus if a justly tuned system is based on the octave and fifths it would contain only two dimensions. W. A. Mathieu uses this perfect fifths and major thirds also (Mathieu, 1997) (see sargam).

Deutsch and Feroe (1981), and Lerdahl and Jackendoff (1983) use a "reductional format" representing pitch relations by "alphabets" or hierarchy of levels such as the chromatic, diatonic, and triadic. Lerdahl's levels include the octave, perfect fifth, major triad, diatonic scale, and the chromatic scale:

According to David Kopp (2002), "Harmonic space, or tonal space as defined by Fred Lerdahl, is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise." (p.1)

The matrices used in the twelve tone technique are not representations of pitch space as nearness nor farness is not indicated, or even possible since one may not move freely about.

Adapted from the Wikipedia article "History of pitch space", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki